I want to make sure I understand the character creation rules. I've never played Traveller, I got a ton of PDFs (and, hopefully soon, some physical books) from backing the Great Rift Kickstarter. I think my notation is pretty clear, but please don't hesitate to ask.

I started by rolling 2d6, six times, and assigning those numbers to the six attributes:

Hannah squeaks past the entrance exams to her homeworld's naval academy, but applies herself in her courses and graduates with honors. Her future seems bright! My first question is this: does the Mechanic 0 she gains from basic training stack with her background skill, giving her Mechanic 1?

I decide that Sublieutenant Cooper is one of the ship's computer-systems officers; her special assignment is overseeing an upgrade of the ship's tactical software. She sticks the landing, and is promoted again. She also seems to be a member of the ship's fencing club. O.o

Disaster! I haven't worked out the nature of the accident, but obviously it wasn't really Lt. Cooper's fault. Another officer, a snot-nosed scion of the aristocracy no doubt, tried to blame her for his own catastrophic lapse in judgment.

Hannah did pretty well for herself over her eight years of service; she leaves the Navy with a decent chunk of change in her pocket, plus a lifetime membership in the Travellers' Aid Society, and enough credit and other assets that she could make a decent down payment on a ship of her own.

It looks like you've forgotten one benefit of graduation. Because she is entering the same career as the academy, three service skills become level 1.

Also, in the first year of a new career, you don't get a skill roll but do basic training instead. However, I find that rule a little confusing when it comes to military academy, as you do the basic training there. So, in my game, I do the same as you've done and allow a skill roll.

In the "Military Academy" section on p.15, it says, "Skills: Gain all Service Skills of the military career the academy is tied to at level 0, as with basic training." I took that to mean that basic training is part of the academy curriculum.